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Concept

Descartes rule of signs

The rule that a polynomial has at most as many positive roots as its coefficients have sign changes, and fewer by an even number. It bounds roots by the number of terms rather than the degree, and it is exact when every root is real.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The proof of Descartes' rule, one term at a time. Four stacked graphs: a four-term polynomial with three positive roots, and the three polynomials obtained by dividing out a power of x and differentiating, each with one fewer term, one fewer sign change and one fewer positive root.

What the signs allow

Descartes' rule reads the signs of a polynomial's coefficients and bounds its positive roots by how often they change — a bound set by the number of terms, not the degree, and proved by nothing more than Rolle's theorem. Each count the rule allows can be had. But not every pair of counts, positive and negative, that the rule allows together can be had together: the first combination that never occurs is at degree four, and the list of impossible ones is still being worked out.

geometry · Euclidean algorithm Two curves with five crossings: x⁶ + (44/31)y³ − y and its mirror. Two curves in the positive quadrant, each the zero set of a trinomial, crossing at five marked points.

Five where four were promised

In one unknown, a polynomial with three terms has at most two positive roots, whatever its degree. The natural guess for two equations in two unknowns, each with three terms, was two times two: four. It is wrong. Bertrand Haas found two such equations in 2002 whose curves cross five times in the positive quadrant, all five crowded within a tenth of one corner, and five turned out to be the most there can ever be.

geometry · Euclidean algorithm

Named alongside it

The objects these essays reach for when they reach for this one.

PolynomialRootsCounterexampleDerivativeExact arithmeticFewnomialLogarithmParityResultantRolles theoremSignSystem of equations

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